i think you can get \(\ddot x(t) = 4 \pi^2 \frac{x}{\sqrt{x^2 + y^2}}\) just by ploughing through mechanically, and you get the same for y, pattern matched, and you'd then need polar to take it further. but i have no idea if that contextually makes any sense. looks like a repulsive radial field, but clutching at straws maybe.